Chemistry Solubility Rule From Jcost Solubility Rule Cert

A certificate, not a law

Solubility rules are the chemist's shorthand for which salts dissolve in water: sodium chloride dissolves, silver chloride does not, and the difference is usually taught as a list to memorize. The Recognition Science declaration SolubilityRuleCert does not add a new rule to that list. It is a certificate, a small bundle of three formally checked facts about a single cost function applied to a ratio of two numbers.

The certificate's subject is a function the framework calls domainCost, defined as J(m / e), where J is the framework's forced cost function and m and e are real numbers. The three facts the certificate proves are general properties of that function, not properties of any chemical system. First, when the two inputs are equal and nonzero, the cost is zero. Second, for positive inputs, the cost is never negative. Third, a constant called the canonical threshold, phi minus 3/2, is positive. These are true for any two positive real numbers, whether they stand for solvation energy and lattice energy, prices, or distances.

The declaration's own docstring is explicit about the limit. It states that the code proves those three general facts and nothing specific to this subject, because domainCost is defined without reference to any chemical quantity. The paragraph describing a solubility rule, where a compound dissolves when J(ΔG_solvation/ΔG_lattice) is below J(φ), is a research note recording where the idea was meant to go, not a result. The certificate itself contains no definition of m or e in chemical terms; without such a definition, it cannot be a theorem about solubility.

What the certificate does establish is a reusable template. The same body of three facts is shared verbatim with 2383 sibling modules, each one a placeholder for a different intended application. The universal content lives in a single module, DomainCostTemplate, and every subject-specific module is meant to supply its own definitions of m and e. SolubilityRuleCert is the chemistry instance of that template, but the chemistry instance is incomplete: it names the intended subject without connecting the certificate's variables to it.

In plain terms, the declaration proves a small piece of mathematics about a cost function and packages it under a chemistry name. The name is a promise of future work, not a description of what the certificate currently does. A reader who wants a solubility rule will not find one here; a reader who wants to see how the framework structures an application will find a clean, honest example of a claim that knows its own limits.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

structure SolubilityRuleCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not establish any empirical solubility rule for any specific salt. The certificate does not connect its cost function to solvation or lattice energies in a formal way. The certificate does not derive the intended threshold J(φ) from chemical principles.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/SolubilityRuleFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: